Years (centuries?) ago when the neighborhood kids had a brief interest in kazoos, we pretty much had to make our own from scratch. Materials were limited, but we found that ordinary kitchen waxed paper was satisfactory if you prepared it properly. You need something as flexible as possible, but the diaphragm shouldn't "damp" the vibrations, so to make the waxed paper work you needed to place it between a couple of pieces of "something absorbent" and run mama's flatiron over it, so that the excess wax got melted and sucked into the "something."

Today, you might use a paper towel for the "sucker" but back then we snuck one of maw's flour sack dish towels out (and carefully refolded it when we put it back, if we hadn't scorched it too badly - in which case we wadded it up and hid it way back under the back porch).

(Modern flatirons have thermostats. Bringing one to proper temperature on a kerosine stove required skill.)

The same rule would apply if you chose to "oil" a piece of parchment or vellum. After the oil is soaked in, hot-pressing it between absorbers to remove as much oil as possible likely will improve the performance.

Both the de-waxed wax paper or oiled vellum should be relatively resistant to absorbing moisture, which is a desirable property for kazoo diaphragms, one would think.

John